Symplectic Rigidity of Real Bidisc

TL;DR

This paper investigates the symplectic rigidity of real bidiscs, proving conditions under which orthogonal transformations preserve symplectic structure and showing certain product domains are not symplectomorphic.

Contribution

It characterizes orthogonal transformations that preserve symplectic structure of bidiscs and establishes non-symplectomorphism between real and complex bidiscs for certain parameters.

Findings

Orthogonal transformations preserving symplectic form are unitary or conjugate to unitary.

Real bidisc and complex bidisc are not symplectomorphic for r ≥ 1.

Certain product domains are distinguished by symplectic properties.

Abstract

Let $\mathbb{D}$ be the unit disc in $\mathbb{C}$, then $\mathbb{D}^n(r)$ is the complex or symplectic $n$-discs of radius $r$. Let $z_j = x_j+iy_j\in\mathbb{C}, j=1,2$ and $\mathbb{D}_{\mathbb{R}}^2=\{(z_1,z_2) : |x_1|^2+|x_2|^2<1,|y_1|^2+|y_2|^2<1\}$ be the real bidisc. In this paper we will prove the following two theorems: 1) If $T\in O(4)$ is an orthogonal transformation on $\mathbb{R}^4$, then $T(\mathbb{D}^2)$ is symplectomorphic to $\mathbb{D}^2$ w.r.t. the standard symplectic form on $\mathbb{R}^4$ if and only if $T$ is unitary or conjugate to unitary. 2) For $r\geq 1$ and $n\geq 2$, $\mathbb{D}_{\mathbb{R}}^2\times \mathbb{D}^{n-2}(r)$ and $\mathbb{D}^2\times \mathbb{D}^{n-2}(r)$ are not symplectomorphic w.r.t. the standard symplectic form on $\mathbb{C}^n$.